Let $M$ be a compact two-dimensional manifold and, $f \in C^{ \infty}(M,\mathbb{ R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{ O}_{ f}=\{ f \circ h \mid h \in \mathcal{ D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{ D}$ on $C^{ \infty}(M,\mathbb{ R})$, and by $\mathcal{ S}(f)=\{ h\in\mathcal{ D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{ S}(f)$ induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{ D}_{ \mathrm{ id}}(M)$ be the identity path component of $\mathcal{ D}(M)$, $\mathcal{ S}'(f)= \mathcal{ S}(f) \cap \mathcal{ D}_{ \mathrm{ id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to $\mathcal{ S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{ O}_{ f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{ 2}$. In this situation $\Gamma_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{ Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{ 2}\to\mathbb{ R}$ whose fixed subtree $\mathrm{ Fix}(G_f)$ consists of more than one point.
Categories: math.GT